Taiwanese Journal of Mathematics

GLOBAL AND NON-GLOBAL SOLUTIONS OF A NONLINEAR PARABOLIC EQUATION

Jong-Shenq Guo, Yung-Jen Lin Guo, and Chi-Jen Wang

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Abstract

We study the global and non-global existence of positive solutions of a nonlinear parabolic equation. For this, we consider the forward and backward self-similar solutions of this equation. We obtain a family of radial symmetric global solutions which tend to zero as the time tends infinity. Next, we show that there are initial data for which the corresponding solutions blow up in finite time. Finally, we also construct some self-similar single-point blow-up patterns with different oscillations.

Article information

Source
Taiwanese J. Math., Volume 9, Number 2 (2005), 187-200.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500407795

Digital Object Identifier
doi:10.11650/twjm/1500407795

Mathematical Reviews number (MathSciNet)
MR2142572

Zentralblatt MATH identifier
1086.34008

Subjects
Primary: 34A12: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions 35B40: Asymptotic behavior of solutions 35K55: Nonlinear parabolic equations

Keywords
nonlinear parabolic equation forward and backward self-similar solutions blow-up patterns

Citation

Guo, Jong-Shenq; Lin Guo, Yung-Jen; Wang, Chi-Jen. GLOBAL AND NON-GLOBAL SOLUTIONS OF A NONLINEAR PARABOLIC EQUATION. Taiwanese J. Math. 9 (2005), no. 2, 187--200. doi:10.11650/twjm/1500407795. https://projecteuclid.org/euclid.twjm/1500407795


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References

  • J. Bebernes and D. Eberly, "Mathematical Problems from Combustion Theory", Applied Mathematical Sciences No. 83, Springer-Verlag, New York, 1989.
  • C. J. Budd and Y.-W. Qi, The existence of bounded solutions of a semilinear elliptic equation, J. Diff. Eq. 82 (1989), 207-218.
  • V. A. Galaktionov and J. L. Vazquez, Continuation of blowup solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math. 50 (1997), 1-67.
  • Y. Giga, On elliptic equations related to self-similar solutions for nonlinear heat equations, Hiroshima Math. J. 16 (1986), 539-552.
  • Y.-J. L. Guo, The forward self-similar equation for a nonlinear parabolic equation, Bulletin of the Institue of Mathematics Academia Sinica 30 (2002), 229-238.
  • J.-S. Guo, Y.-J. L. Guo, and J.-C. Tsai, Single-point blow-up patterns for a nonlinear parabolic equation, Nonlinear Analysis 53 (2003), 1149-1165.
  • S. Kaplan, On the growth of solutions of quasilinear parabolic equations, Comm. Pure Appl. Math. 16 (1957), 305-330.
  • Y.-W. Qi, The self-similar solutions to a fast diffusion equation, Z. Angew. Math. Phys. 45 (1994), 914-932.
  • A. A. Samarskii, V. A. Galaktionov, S.P. Kurdyumov, and A.P. Mikhailov, "Blow-up in Quasilinear Parabolic Equations", de Gruyter Exposition in Mathematics No. 19, Walter de Gruyter, Berlin, 1995.
  • W. C. Troy, The existence of bounded solutions of a semilinear heat equation, SIAM J. Math. Anal. 18 (1987), 332-336.